A Class of Operators with Symbol on the Bloch Space of a Bounded Homogeneous Domain




Allen, Robert Francis

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<pre>Let X be a Banach space of holomorphic functions on a domain D in C^n. If ψ is a holomorphic function on D, and ϕ is a holomorphic self-map of D, we defi ne the weighted composition operator on X with symbols ψ and ϕby W ψ,ϕf = (f о ϕ). This operator is a generalization of the multiplication operator M(ψ) f = ψ f and the composition operator Cϕf = f о ϕ, which are known as degenerate weighted composition operators. The weighted composition operators have been an object of interest since the early 30's with their connection to the isometries of various spaces of analytic functions on the unit disk. The Bloch space has been of interest since the early 70's to function theorists and to operator theorists. However, these two concepts did not meet until 2001. Classical operator theory on spaces of holomorphic functions in several complex variables is typically carried out on the unit ball and the unit polydisk. The respective function theories are very di fferent. In this dissertation, we attempt to unify the operator theory on the Bloch space on these domains and extend it further to bounded homogeneous domains in C^n. In this uni ed manner, we study the fundamental properties of the weighted composition operators: 1. For what symbols ψ and ϕ is W ψ,ϕ bounded? 2. For what symbols ψ and ϕ is W ψ,ϕ compact? 3. What is an expression for ||W ψ,ϕ||? 4. For what symbols ψ and ϕ is Wψ,ϕ an isometry? 5. What is the spectrum of Wψϕ? It is our hope that this work will mark the beginning of a paradigm shift in operator theory research in several complex variables. This will bring in new fields of study such as diff erential geometry into the study of operators, thus enriching the field. </pre>



Weighted Composition Operators, Bloch Space, Homogeneous Domain, Operator Theory